Williams was born in Worcester on 28 April 1828, the son of a civil engineer also named Edward (responsible for works to make the River Severn navigable; also a keen amateur artist and friend of John Constable) and his Quaker wife Sarah Whiting. His brother Benjamin Williams Leader became a famous landscape artist. The family lived at Diglis House in Worcester (today the Diglis Hotel).
After attending the Royal Grammar School Worcester Williams became an apprentice to his father. In 1856 he was chosen out of 110 applicants to be chief engineer for the development of the navigable northern section of the River Weaver in Cheshire. He began to specialise in canal construction and in 1865 produced plans for enlarging the Weston Canal, a short stretch of canal linking the river near Frodsham to docks at Weston Point, Runcorn.
On 1 September 1872, he joined the Bridgewater Navigation Company and worked on the Bridgewater Canal.
After submitting proposals in competition with another engineer (Hamilton Fulton), Williams was then appointed by Daniel Adamson in 1882 to design a new ship canal linking Manchester with the Irish Sea. Williams became chief designer and chief engineer, helping the Manchester Ship Canal Company formulate its proposals for the necessary Act of Parliament. From its initial reading in 1883, it took two years for the Bill to receive Royal Assent, and a further two years before the first construction work started, in November 1887.
The canal opened in 1894, and has been described as "a feat without precedent in modern history". As well as the canal itself, major engineering landmarks of the scheme included the Barton Swing Aqueduct (carrying the Bridgewater Canal over the Ship Canal) and a neighbouring swing bridge for road traffic at Barton.